## Wednesday, 16 December 2020

### Energy conditions

The equation of state of the prefect fluid that models the Friedmann universe is $$p=w\rho$$where $p,\rho$ are pressure and energy density and $w$ is a constant. In section 4.6 Carroll describes five 'popular' energy conditions on the energy-momentum tensor without saying why they are popular or even why they are interesting and shows that for any of them to be true we must have $w\geq-1$ in the equation of state. The energy conditions, which sound like characters in a robot movie, are WEC, NEC, DEC, NDEC and SEC. Possible values of $\rho,p$ are shown in the diagram below for these energy conditions along with possible values when $w\geq-1$ in the equation of state. Assuming that our Friedmann universe has $\rho>0$ (highly plausible) and satisfies one of the robots (haven't got the faintest idea) then we must have $w\geq-1$.

From the equation of state $w\geq-1$ gives$$\frac{p}{\rho}\geq-1\Rightarrow p\geq-\rho\Rightarrow p+\rho\geq0$$which is the same as the NEC. So why aren't the pictures (b) and (f) the same? I hope you've seen my error. Luckily I did and have avoided falsely accusing Carroll of a slip up.

Here is my limited understanding: Commentary 4.6 Energy conditions.pdf (3 pages)